L(s) = 1 | + i·2-s − 4-s − 5-s − i·8-s − i·10-s + 1.53i·11-s + 1.48i·13-s + 16-s + 2.42·17-s − 4.86i·19-s + 20-s − 1.53·22-s + 0.267i·23-s + 25-s − 1.48·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.447·5-s − 0.353i·8-s − 0.316i·10-s + 0.461i·11-s + 0.411i·13-s + 0.250·16-s + 0.589·17-s − 1.11i·19-s + 0.223·20-s − 0.326·22-s + 0.0558i·23-s + 0.200·25-s − 0.290·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.480697712\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480697712\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.53iT - 11T^{2} \) |
| 13 | \( 1 - 1.48iT - 13T^{2} \) |
| 17 | \( 1 - 2.42T + 17T^{2} \) |
| 19 | \( 1 + 4.86iT - 19T^{2} \) |
| 23 | \( 1 - 0.267iT - 23T^{2} \) |
| 29 | \( 1 - 0.898iT - 29T^{2} \) |
| 31 | \( 1 + 0.828iT - 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 + 3.47iT - 53T^{2} \) |
| 59 | \( 1 - 6.25T + 59T^{2} \) |
| 61 | \( 1 + 6.62iT - 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 0.343iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 14.9iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.387651648553166984906353590251, −7.72402169986798186136660167738, −7.02539272481907059452745839056, −6.54157725705956882832151906154, −5.52335113173945982189613384282, −4.87723944359507554078759838854, −4.12353167718114816500448409423, −3.28853741683657045122283656134, −2.16035108583639653070772360550, −0.78235254891509976209709504911,
0.61268540996737037462001161984, 1.69110606812364556543228110934, 2.82933041631911591740626891706, 3.55216761696990545590474428097, 4.21300816981134847185551113353, 5.22740284089959239242580729428, 5.83161089149655703349215776001, 6.76243133623960739823949399361, 7.79776413543172654408646178811, 8.124479483257670720790669054222